Categorical quantum-group realization of the A-twisted theory

Fix integers $n\geq 2$ and $k\geq n$, set $q=e^{i\pi/k}$, and let $\mathcal C^{(n,k)}$ be the coherent sheaf of dg categories of line operators of $\mathcal T_{n,k}^A$ over $PGL(n,\mathbb C)$. Quantum-group TQFT conjecture. There is an equivalence

$$\mathcal C^{(n,k)}\simeq D^b\big(U_q(\mathfrak{sl}_n)\text{-mod}\big),$$

where the quantum group is the simply connected De Concini–Kac quantum group at an even root of unity, with Frobenius center acting semisimply. More generally, $\mathcal T_{n,k}^A$ should define the stated extended axiomatic TQFT and agree in cohomological degree zero with the CGP TQFT based on $U_q^H(\mathfrak{sl}_n)$. The paper supplies proofs and computational evidence for special cases, including $n=2$ at trivial background holonomy, but the general equivalence remains conjectural.